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Properties

Label 2-630-63.25-c1-0-22
Degree $2$
Conductor $630$
Sign $-0.124 + 0.992i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (1.21 − 1.23i)3-s + 4-s + (0.5 − 0.866i)5-s + (−1.21 + 1.23i)6-s + (−1.09 + 2.40i)7-s − 8-s + (−0.0482 − 2.99i)9-s + (−0.5 + 0.866i)10-s + (−0.554 − 0.960i)11-s + (1.21 − 1.23i)12-s + (−1.64 − 2.84i)13-s + (1.09 − 2.40i)14-s + (−0.461 − 1.66i)15-s + 16-s + (2.19 − 3.80i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.701 − 0.712i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.495 + 0.504i)6-s + (−0.414 + 0.910i)7-s − 0.353·8-s + (−0.0160 − 0.999i)9-s + (−0.158 + 0.273i)10-s + (−0.167 − 0.289i)11-s + (0.350 − 0.356i)12-s + (−0.455 − 0.788i)13-s + (0.292 − 0.643i)14-s + (−0.119 − 0.431i)15-s + 0.250·16-s + (0.532 − 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.124 + 0.992i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801689 - 0.909014i\)
\(L(\frac12)\) \(\approx\) \(0.801689 - 0.909014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.21 + 1.23i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.09 - 2.40i)T \)
good11 \( 1 + (0.554 + 0.960i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.64 + 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.19 + 3.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.622 + 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.14 + 5.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.26 + 5.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.10T + 31T^{2} \)
37 \( 1 + (-0.659 - 1.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.60 - 6.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.88 - 8.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (-2.73 + 4.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.46T + 59T^{2} \)
61 \( 1 - 4.40T + 61T^{2} \)
67 \( 1 + 5.23T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + (0.731 - 1.26i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.41T + 79T^{2} \)
83 \( 1 + (4.06 - 7.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.42 + 2.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.54 - 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.847468808106683104547834039952, −9.568261038265502533741354219232, −8.360057356716816650586045993380, −8.157934905585564057174533529954, −6.87767180042275310182654449353, −6.12916601433409887650464196467, −4.96537868363220530619066548152, −3.05968718923128761716192659330, −2.43552873561276168320235241440, −0.77336241069871618124826443583, 1.76487442993915784144351759858, 3.12964714264948077993611842728, 4.02567645311604406252208324054, 5.30725400994546749401925326955, 6.71000606838324964437479429668, 7.39443704749702914868072937604, 8.316751237897928151326825042207, 9.262706134796356481392074008698, 9.981629890156868671665350188553, 10.46129162057960176064459736293

Graph of the $Z$-function along the critical line